Kirill M. Kuzanyan

Magnetic helicity evolution during the solar activity cycle: Observations and dynamo theory

We study a simple model for the solar dynamo in the framework of the Parker migratory dynamo, with a nonlinear dynamo saturation mechanism based on magnetic helicity conservation arguments. We find a parameter range in which the model demonstrates a cyclic behaviour with properties similar to that of Parker dynamo with the simplest form of algebraic α -quenching. We compare the nonlinear current helicity evolution in this model with data for the current helicity evolution obtained during 10 years of observations at the Huairou Solar Station of China. On one hand, our simulated data demonstrate behaviour comparable with the observed phenomenology, provided that a suitable set of governing dynamo parameters is chosen. On the other hand, the observational data are shown to be rich enough to reject some other sets of governing parameters. We conclude that, in spite of the very preliminary state of the observations and the crude nature of the model, the idea of using observational data to constrain our ideas concerning magnetic field generation in the framework of the solar dynamo appears promising.

The radial distribution of magnetic helicity in the solar convective zone: observations and dynamo theory

We continue our attempt to connect observational data on current helicity in solar active regions with solar dynamo models. In addition to our previous results about temporal and latitudinal distributions of current helicity, we argue that some information concerning the radial profile of the current helicity averaged over time, and latitude can be extracted from the available observations. The main feature of this distribution can be presented as follows. Both shallow and deep active regions demonstrate a clear dominance of one sign of current helicity in a given hemisphere during the whole cycle. Broadly speaking, current helicity has opposite polarities in the Northern and Southern hemispheres, although there are some active regions that violate this polarity rule. The relative number of active regions violating the polarity rule is significantly higher for deeper active regions. A separation of active regions into ‘shallow’, ‘middle’ and ‘deep’ is made by comparing their rotation rate and the helioseismic rotation law. We use a version of Parker’s dynamo model in two spatial dimensions, which employs a nonlinearity based on magnetic helicity conservation arguments. The predictions of this model about the radial distribution of solar current helicity appear to be in remarkable agreement with the available observational data; in particular the relative volume occupied by the current helicity of ‘wrong’ sign grows significantly with the depth.

A dynamo wave in an inhomogeneous medium

Abstract The Maximally-Efficient Generation Approach (MEGA) for an αω-dynamo is investigated based on the simplest model of the Parker Migratory Dynamo. The α-effect is supposed inhomogeneous. The equations which govern the magnetic field are solved with the use of a WKB method. Turning points and the behavior of the solution in their vicinity are derived. The point of maximum dynamo wave amplitude is shown to be shifted from the point where the generation sources are maximum towards the direction of the dynamo wave propagation. For the solar dynamo the obtained direction of the dynamo wave propagation reverses over the subpolar domains. Dynamo waves with a poleward migration are observable over the polar regions of the Sun. The generation threshold is calculated. The properties of the dynamo waves are compared with observational data for the solar cycle. An asymptotic solution for oscillating modes of the galactic disk dynamo is also obtained.

Linear theory of compressible convection in rapidly rotating spherical shells, using the anelastic approximation

The onset of compressible convection in rapidly rotating spherical shells is studied in the anelastic approximation. An asymptotic theory valid at low Ekman number is developed and compared with numerical solutions of the full equations. Compressibility is measured by the number of density scale heights in the shell. In the Boussinesq problem, the location of the onset of convection is close to the tangent cylinder when there is no internal heating only a heat flux emerging from below. Compressibility strongly affects this result. With only a few scale heights or more of density present, there is onset of convection near the outer shell. Compressibility also strongly affects the frequencies and preferred azimuthal wavenumbers at onset. Compressible convection, like Boussinesq convection, shows strong spiralling in the equatorial plane at low Prandtl number. We also explore how higher-order linear modes penetrate inside the tangent cylinder at higher Rayleigh numbers and compare modes both symmetric and antisymmetric about the equator.

A new dynamo pattern revealed by solar helical magnetic fields

A previously unobservable mirror asymmetry of the solar magnetic field – a key ingredient of the dynamo mechanism which is believed to drive the 11-year activity cycle – has recently been measured. This was achieved through systematic monitoring of solar active regions carried out for more than 20 years at observatories in Mees, Huairou and Mitaka. In this Letter we report on detailed analysis of vector magnetic field data, obtained at Huairou Solar Observing Station in China. Electric current helicity (the product of current and magnetic field components in the same direction) was estimated from the data and a latitude–time plot of solar helicity during the last two solar cycles has been produced. We find that like sunspots helicity patterns propagate equatorwards, but unlike sunspot polarity helicity in each solar hemisphere does not change sign from cycle to cycle, thus confirming the theory. There are, however, two significant time–latitudinal domains in each cycle when the sign briefly inverts. Our findings shed new light on stellar and planetary dynamos and are yet to be included in the theory.

A nonlinear dynamo wave riding on a spatially varying background

A systematic asymptotic investigation of a pair of coupled nonlinear one–dimensional amplitude equations, which provide a simplified model of solar and stellar magnetic activity cycles, is presented. Specifically, an αΩ–dynamo in a thin shell of small gap–to–radius ratio &b.epsi; (≪ 1) is considered, in which the Ω–effect (the differential rotation) is prescribed but the α–effect is quenched by the finite–amplitude magnetic field. The unquenched system is characterized by a latitudinally θ–dependent dynamo number D, with a symmetric single–hump profile, which vanishes at both the pole, θ = π/2, and the equator, θ = 0, and has a maximum, D, at mid–latitude, θM = π/4. The shape D(θ)/D is fixed, so that there is only a single driving parameter D. At onset of global instability, D = DL(ϵ): = DT + O(ϵ), a travelling wave, of frequency ω = ωL(ϵ): = ωT+ O(ϵ) and wavelength O(ϵ), is localized at a low latitude θPT(< θM); DT and ωT are constants independent of ϵ. As a consequence of the spatial separation of θPT and θM, the squared field amplitude increases linearly with the excess dynamo number D – DL in the weakly nonlinear regime, as usual, but with a large constant of proportionality dependent on some numerically small power of exp(1/ϵ). Whether the bifurcation is sub– or supercritical is extremely sensitive to the value of ϵ. In the nonlinear regime, the travelling wave localized at θPT at global onset expands and lies under an asymmetric envelope that vanishes smoothly at a low latitude θP but terminates abruptly on a length O(ϵ) – comparable to the wavelength – across a front at high latitude θF. The criterion of Dee and Langer, applied to the local linear evanescent disturbance ahead of the front, determines the lowest order value of the frequency close to the global onset value ωT. The global transition is characterized by the abrupt shift of θF from θPT to θM; during that passage, D executes O(ϵ-1) oscillations of increasing magnitude about DL. Fully developed nonlinearity occurs when θF > θM. In that regime, Meunier and co–workers showed that the O(1) quantities θF – θM and (ω – ωT)/ϵ2/3 increase together in concert with D – DT. By analysing the detailed structure of the front of width O(ϵ), we obtain ω correct to the higher order O(ϵ) and show improved agreement with numerical integrations performed by Meunier and co–workers of the complete governing equations at finite ϵ.

Current Helicity of Active Regions as a Tracer of Large-scale Solar Magnetic Helicity

We demonstrate that the current helicity observed in solar active regions traces the magnetic helicity of the large-scale dynamo generated field. We use an advanced two-dimensional mean-field dynamo model with dynamo saturation based on the evolution of the magnetic helicity and algebraic quenching. For comparison, we also studied a more basic two-dimensional mean-field dynamo model with simple algebraic alpha-quenching only. Using these numerical models we obtained butterfly diagrams both for the small-scale current helicity and also for the large-scale magnetic helicity, and compared them with the butterfly diagram for the current helicity in active regions obtained from observations. This comparison shows that the current helicity of active regions, as estimated by −A · B evaluated at the depth from which the active region arises, resembles the observational data much better than the small-scale current helicity calculated directly from the helicity evolution equation. Here B and A are, respectively, the dynamo generated mean magnetic field and its vector potential. A theoretical interpretation of these results is given.

The distribution of current helicity at the solar surface at the beginning of the solar cycle

A fraction of solar active regions are observed to have current helicity of a sign that contradicts the polarity law for magnetic helicity; this law corresponds to the well-known polarity law for sunspots. A significant excess of active regions with the “wrong” sign of helicity is seen to occur just at the beginning of the cycle. We compare these observations with predictions from a dynamo model based on principles of helicity conservation, discussed by Zhang et al. (2006). This model seems capable of explaining only a fraction of the regions with the wrong sign of the helicity. We attribute the remaining excess to additional current helicity production from the twisting of rising magnetic flux tubes, as suggested by Choudhuri et al. (2004a). We estimate the relative contributions of this effect and that connected with the model based on magnetic helicity conservation. c

Probing signatures of the alpha-effect in the solar convection zone

An attempt to extract maximum information on signatures of the alpha-effect from current helicity and twist density calculations in the solar photosphere is carried out. A possible interpretation of the results for developing the dynamo theory is discussed. The analysis shows that the surface magnetic current helicity is mainly negative/positive in the northern/southern hemispheres of the Sun. This indicates the actual alpha-effect at the photospheric level to be positive/negative, respectively. However, at the bottom of the convection zone, we may assume this effect to change the sign to negative/positive. We reveal some quantities related to the alpha-effect and discuss its spatial and temporal distribution. It is also found that there are a small number of active regions where the sign of the alpha-effect is opposite to that in most active regions. Such exceptional active regions seem to localize at certain active longitudes. We compare the determined regularities with theoretical predictions of the alpha-effect distribution in the solar convection zone.

HALF-WIDTH OF A SOLAR DYNAMO WAVE IN PARKER'S MIGRATORY DYNAMO

A kinematic αω-dynamo model of magnetic field generation in a thin convection shell with nonuniform helicity for large dynamo numbers is considered in the framework of Parkers migratory dynamo. The asymptotic solution obtained of equations governing the magnetic field has the form of an anharmonic travelling dynamo wave. This wave propagates over most latitudes of the solar hemisphere from high latitudes to the equator, and the amplitude of the magnetic field first increases and then decreases with propagation. Over the subpolar latitudes, the dynamo wave reverses; there the dynamo wave propagates polewards and decays with latitude. The half-width of the maximum of the magnetic field localisation and the phase velocity of the dynamo wave are calculated. Butterfly diagrams are plotted and analysed and these show that even a simple model may reveal some properties of the solar magnetic fields.